# Embedding topological spaces in a type of generalized topological spaces

Document Type : Original Article

Author

Department of Mathematics, Payame Noor University, P.O. Box, 19395-3697, Tehran, Iran.

10.22034/kjm.2020.109822

Abstract

A stack on a nonempty set $X$ is a collection of nonempty subsets of $X$ that is closed under the operation of superset.
Let $(X, \tau)$ be an arbitrary topological space with a stack $\mathcal{S}$, and let $X^*=X \cup \{p\}$ for $p \notin X$. In the present paper, using the stack $\mathcal{S}$ and the topological closure operator associated  to the space $(X, \tau)$, we define an envelope operator on $X^*$ to construct a generalized topology $\mu_\mathcal{S}$ on $X^*$.
We then show that the space $(X^*, \mu_\mathcal{S})$ is the generalized extension of the space $(X, \tau)$. We also provide  conditions under which $(X^*, \mu_\mathcal{S})$ becomes a generalized Hausdorff space.

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